Sharp, quantitative bounds on the distance between a polynomial piece and its Bezier control polygon

The maximal distance between a Bezier segment and its control polygon is bo unded in terms of the differences of the control point sequence and a const ant that depends only on the degree of the polynomial. The constants derive d here for various norms and orders of differences are the smallest possibl e. In particular, the bound in terms of the maximal absolute second difference of the control points is a sharp upper bound for the Hausdorff distance be tween the control polygon and the curve segment. It provides a straightforw ard proof of quadratic convergence of the sequence of control polygons to t he Bezier segment under subdivision or degree-fold degree-raising, and esta blishes the explicit convergence constants, and allows analyzing the optima l choice of the subdivision parameter for adaptive refinement of quadratic and cubic segments and yields efficient bounding regions. (C) 1999 Elsevier Science B.V. All rights reserved.